New Results on Codes with Covering Radius 1 and Minimum Distance 2

2005 ◽  
Vol 35 (2) ◽  
pp. 241-250 ◽  
Author(s):  
Patric R. J. �sterg�rd ◽  
J�rn Quistorff ◽  
Alfred Wassermann
Keyword(s):  
Minimum Distance ◽  
Covering Radius

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals Bounds for the Minimum Distance and Covering Radius of Orthogonal Arrays via Their Distance Distributions

2021 ◽  
Author(s):  
Silvia Boumova ◽  
Peter Boyvalenkov ◽  
Maya Stoyanova
Keyword(s):  
Minimum Distance ◽  
Orthogonal Arrays ◽  
Analytic Form ◽  
Covering Radius ◽  
Distance Distributions ◽  
Feasible Sets ◽  
Ongoing Project

We propose two methods for obtaining estimations on the minimum distance and covering radius of orthogonal arrays. Both methods are based on knowledge about the (feasible) sets of distance distributions of orthogonal arrays with given length, cardinality, factors and strength. New bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of orthogonal arrays with parameters in doable ranges.

The covering radius of the Leech lattice

1982 ◽  
Vol 380 (1779) ◽  
pp. 261-290 ◽  
Keyword(s):  
Minimum Distance ◽  
Dimensional Space ◽  
Covering Radius ◽  
Lattice Points ◽  
Maximum Distance ◽  
Leech Lattice

We investigate the points in 24-dimensional space at maximum distance from the Leech lattice, i. e. the ‘deepest holes’ in that lattice. The maximum distance of any such point from the Leech lattice is shown to be 1/√2 times the minimum distance between the lattice points. Furthermore there are 23 types of ‘deepest hole’, one for each of the 23 even unimodular 24-dimensional lattices found by Niemeier.

scholarly journals On the weight distribution of the cosets of MDS codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Weight Distribution ◽  
Minimum Distance ◽  
Covering Radius ◽  
Mds Codes ◽  
Mds Code ◽  
Weight Distributions ◽  
Maximum Distance Separable ◽  
Special Relations ◽  
Saturating Set

<p style='text-indent:20px;'>The weight distribution of the cosets of maximum distance separable (MDS) codes is considered. In 1990, P.G. Bonneau proposed a relation to obtain the full weight distribution of a coset of an MDS code with minimum distance <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> using the known numbers of vectors of weights <inline-formula><tex-math id="M2">\begin{document}$ \le d-2 $\end{document}</tex-math></inline-formula> in this coset. In this paper, the Bonneau formula is transformed into a more structured and convenient form. The new version of the formula allows to consider effectively cosets of distinct weights <inline-formula><tex-math id="M3">\begin{document}$ W $\end{document}</tex-math></inline-formula>. (The weight <inline-formula><tex-math id="M4">\begin{document}$ W $\end{document}</tex-math></inline-formula> of a coset is the smallest Hamming weight of any vector in the coset.) For each of the considered <inline-formula><tex-math id="M5">\begin{document}$ W $\end{document}</tex-math></inline-formula> or regions of <inline-formula><tex-math id="M6">\begin{document}$ W $\end{document}</tex-math></inline-formula>, special relations more simple than the general ones are obtained. For the MDS code cosets of weight <inline-formula><tex-math id="M7">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> and weight <inline-formula><tex-math id="M8">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> we obtain formulas of the weight distributions depending only on the code parameters. This proves that all the cosets of weight <inline-formula><tex-math id="M9">\begin{document}$ W = 1 $\end{document}</tex-math></inline-formula> (as well as <inline-formula><tex-math id="M10">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula>) have the same weight distribution. The cosets of weight <inline-formula><tex-math id="M11">\begin{document}$ W = 2 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ W = d-2 $\end{document}</tex-math></inline-formula> may have different weight distributions; in this case, we proved that the distributions are symmetrical in some sense. The weight distributions of the cosets of MDS codes corresponding to arcs in the projective plane <inline-formula><tex-math id="M13">\begin{document}$ \mathrm{PG}(2,q) $\end{document}</tex-math></inline-formula> are also considered. For MDS codes of covering radius <inline-formula><tex-math id="M14">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> we obtain the number of the weight <inline-formula><tex-math id="M15">\begin{document}$ W = d-1 $\end{document}</tex-math></inline-formula> cosets and their weight distribution that gives rise to a certain classification of the so-called deep holes. We show that any MDS code of covering radius <inline-formula><tex-math id="M16">\begin{document}$ R = d-1 $\end{document}</tex-math></inline-formula> is an almost perfect multiple covering of the farthest-off points (deep holes); moreover, it corresponds to an optimal multiple saturating set in the projective space <inline-formula><tex-math id="M17">\begin{document}$ \mathrm{PG}(N,q) $\end{document}</tex-math></inline-formula>.</p>

scholarly journals On codes with covering radius 1 and minimum distance 2

2001 ◽  
Vol 12 (4) ◽  
pp. 449-452 ◽  
Author(s):  
A. Blokhuis ◽  
S. Egner ◽  
H.D.L. Hollmann ◽  
J.H. van Lint
Keyword(s):  
Minimum Distance ◽  
Covering Radius

scholarly journals Graphs Associated with Codes of Covering Radius 1 and Minimum Distance 2

10.37236/792 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Joanne L. Hall
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Latin Squares ◽  
Covering Radius ◽  
Unique Graph ◽  
Partial Latin Squares

The search for codes of covering radius $1$ led Östergård, Quistorff and Wassermann to the OQW method of associating a unique graph to each code. We present results on the structure and existence of OQW-associated graphs. These are used to find an upper bound on the size of a ball of radius $1$ around a code of length $3$ and minimum distance $2$. OQW-associated graphs and non-extendable partial Latin squares are used to catalogue codes of length $3$ over $4$ symbols with covering radius $1$ and minimum distance $2$.

Determining The Minimum Distance Between Centers of Two Parallel Tunnels to Apply The Law of Super Position in Order to Calculate Subsidence by Using The Software FLAC 3D

2015 ◽  
Vol 1 (1) ◽  
pp. 13-20
Author(s):  
Hamid Reza Samadi ◽  
Mohammad Reza Samadi
Keyword(s):  
Minimum Distance ◽  
Surface Subsidence ◽  
Urban Traffic ◽  
Underground Structures ◽  
Rapid Population Growth ◽  
The North ◽  
Flac 3D ◽  
The Law ◽  
3D Software ◽  
Twin Tunnels

Due to the development of cities as well as rapid population growth, urban traffic is increasing nowadays. Hence, to improve traffic flow, underground structures such as metro, especially in metropolises, are inevitable. This paper is a research on the twin tunnels Of Isfahan's metro between Shariaty station and Azadi station from the North towards the South. In this study, simultaneous drilling of subway's twin tunnels is simulated by means of Finite Difference Method (FDM) and FLAC 3D software. Moreover, the lowest distance between two tunnels is determined in a way that the Law of Super Position could be utilized to manually calculate the amount of surface subsidence, resulted by drilling two tunnels, by employing the results of the analysis of single tunnels without using simultaneous examination and simulation. In this paper, this distance is called "effective distance". For this purpose, first, the optimum dimensions of the model is chosen and then, five models with optimum dimensions will be analyzed separately, each of which in three steps. The results of analyses shows that the proportions (L/D) greater than or equal 2.80, the Law of Super Position can be applied for prediction of surface subsidence, caused by twin tunnels' construction

ON R-LINEAR CODES AND ITS COVERING RADIUS OF SOME CLASSES OF LINEAR CODES IN R

2020 ◽  
Vol 9 (3) ◽  
pp. 1159-1176
Author(s):  
P. Chella Pandian ◽  
J. Mahalakshmi
Keyword(s):  
Linear Codes ◽  
Covering Radius
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